For this answer please understand: A domino if you are not familiar with
it is a little wooden or plastic (usually) tile usually about 3 cm by 6 cm
by .75 cm thick. These can be lined up on edge in a row: |||||||||||| and the
first one
can be pushed over: /||||||||||| and this causes the next to fall and so
on: //////|||||||| until eventually they have all fallen
over: //////////////////_.

This explanation: I have broken it into 3 parts.

Sound propagates faster than your (typical) moving object and is the
result of typically hundreds to thousands of "pushes" or oscillations per
second. Sound is also a decoupled phenomenon: it is the result of many
independent air molecules bumping into one another in a coherent way due
to these pressure oscillations at the source.

To set this up in what I hope is an instructive scale analogy: Part 1 is to
consider a snail that knocks over the first of a series of dominoes. The
dominoes are like air molecules. The chain of dominoes moves away from
the snail very fast (from the snail's point of view). The dominoes as
they fall don't "know" about the snail; each domino just knows it got
bumped into by another domino and that it is falling over to bump into a
different domino.

That's part 1. Part 2 is that perceived sound is actually many domino
falls or ripples in rapid succession. So imagine that each domino in your
chain of dominoes can instantly "stand up again" after falling over, perhaps by
being attached somewhere by a rubber band. This
is getting closer to the case of air molecules: They are still there and
can continue bouncing into one another. Now your snail knocks over the
first domino. That sets off a ripple of falling dominoes but that first
domino bounces right back, as do the others. The snail waits a little bit
and knocks the first domino over again. The same thing happens: The
dominoes fall and stand back again. Many ripples. The snail does this for
awhile always waiting the same amount of time before the next push. The
successive ripples of falling dominoes has a frequency, which in terms of
sound is a pitch that we hear. If the snail starts pushing the first
domino over more frequently (waits less time between pushes), by analogy
this is a sound wave of a higher frequency so the perceived pitch goes up.

Notice that if the dominoes were all stacked together, physically
touching one another face to face, and if the speed of sound were just
the last domino moving forwards (rather than repeated waves of falling
dominoes) then the snail's speed *would* determine the speed of sound. The
snail would push all the dominoes at once and the end domino would move
just as fast as the snail. But this is not how sound works in air.

Now for part 3 let's abandon the snails and dominoes in favor of a siren
on an ambulance. The ambulance is rushing along at 50 km / hour and the
siren is going off very loudly. This siren is vibrating the air at say
2000 times per second, so at the speed of sound (about 300 meters per
second) these pressure waves are spreading out like ripples on a pond,
2000 ripples per second from the point of view of the ambulance driver.
These ripples travel by means of air molecules bouncing into one another;
but each air molecule doesn't know about the ambulance. It only knows
that it got bumped into again and again at some frequency, and that it in
turn is bumping into other air molecules at the same frequency. All
together all of these jiggling air molecules create little pressure waves
spaced about 15 cm apart (300 meters per second divided by 2000 waves per
second).

Now suppose you are standing on the sidewalk up ahead of the ambulance.
The ambulance is moving at 50 km per hour (4 meters per second) towards you. So
each successive pressure
wave in the direction the ambulance is moving is created about 2
millimeters closer to you than the previous one (compared to if the ambulance
were not moving). So the pressure waves are
hitting your ears closer together in time than if the ambulance were not
moving. This causes the perceived frequency of the ambulance to become
higher. So the speed of sound does not change but the frequency of the
siren does ahead of the ambulance. As the ambulance passes by it is now
getting further away and the effect is reversed: The frequency of
pressure waves drops and the pitch of the siren becomes lower behind the
ambulance. This curious change in pitch as the ambulance passes by is
often referred to as the Doppler effect.